A Study Of Critical Conditions For Pest-natural Enemy Systems With Nonlinear Impulse Control

It is reasonable and accurate for impulsive differential equations to mathemati-cal simulate the evolution of biological behaviors and complex biological phenomena when the motion states short-term rapidly change their values,which provides con-ditions for people to assist in the design of IPM strategies and understand of the biological phenomena from a mathematical point of view,so as to guide people to know about how to control some target pest outbreaks.Since the solutions of the impulsive systems are discontinuous at the impulsive time point,which make the theories of the impulsive differential systems more complicated and more difficult for people to apply the corresponding theory to solve practical problems.Therefore,it is great theoretical value and practical significance to apply the theories of impulsive differential systems in population dynamics.In recent years,the impulsive differ-ential systems with integrated pest management has been systematically studied and greatly developed,which enriched its basic theory and analytical techniques of impulsive differential system.However,the major assumption in previous studies is that a proportion of the pest population will be killed instantly after spraying pes-ticide,while simultaneously releasing a constant number of natural enemies,which means that the agricultural resources have almost no effect on IPM.In reality,the pest control strategies will inevitably be affected by the limitation of agricultural resources because of the unbalance development of agricultural,and its influence is often nonlinear.It is easy to classify and monitor the real number of pests and natural enemies in the field by modern technology and advanced detection instru-ments,which can help to design the nonlinear impulse control strategies,i.e.the instantaneous releasing numbers of natural enemies should depend on the densities of both pest and natural enemy densities in the field.In order to better simulate the effect of resource limitation,a saturation function of the limited resources is taken into account pest control strategies.The main purpose of this.paper is to construct a predator-prey model with nonlinear impulsive control,and develop the analytical techniques to investigate the effect of limited predator releases on the outbreak of pest populations,so as to reveal important biological conclusions by qualitative analysis and numerical simulations.Meanwhile,the corresponding bi-ological conclusions can provide certain guidance for the agricultural ecologists to find the optimal pest control strategies.In order to describe how the resource limitation affect the pest control strategy,a generalized predator-prey model with nonlinear impulsive control strategy was introduced in chapter 2.The existence and global stability of the pest free periodic solution have been addressed by using Floquet theory and analytical techniques.When the trivial periodic solution loses its stability and a stable nontrivial periodic solution emerges once a threshold condition is reached.We choose the Holling Type? functional response function as an example with aims to investigate how the nonlinear pulse perturbations affect the pest control,and the numerical analysis of the system parameters showed that the dynamic behavior of the model was very complex.In reality,the instantaneous releasing numbers of natural enemies should de-pend on the densities of both pest and natural enemy densities in the field.Based on the above and the models developed in chapter 2,we propose a generalized predator-prey model with nonlinear pulse in chapter 3.The threshold condition for the existence and stability of the pest free periodic solution were obtained,and the condition for the permanence is also given.It is then shown that once a threshold condition is reached,a nontrivial periodic solution appears via a supercritical bifur-cation by employing an operator theoretic approach.Finally,numerical simulations show that the model with Holling Type II functional response function has very com-plex dynamical behavior,including period-doubling bifurcation,chaotic solutions,chaos crisis,period-halving bifurcations,and periodic windows.Moreover,there ex-ists an interesting phenomenon that period-doubling bifurcation and period-halving bifurcation are always duality coexist when nonlinear impulsive controls are taken into IPM strategy,which makes the dynamical behavior of model more complicated and result in more difficult for successful pest control.The aim of IPM should reduce pest populations to below the economic thresh-old rather than eradication,which can be naturally and accurately described by the state dependent impulsive differential equations.In chapter 4,a predator-prey model with nonlinear state-dependent feedback control is proposed and studied.Ac-cording to the position relationship between threshold ET and the equilibria point,the exact domains of impulsive and phase sets are defined and discussed,and the ex-istence and global stability of the order-1 periodic solutions are studied by Poincare map under various cases.The existence and stability of the semi-trivial periodic solution are provided by the analogue of the analogue of Poincare criterion,and the transcritical bifurcation is discussed in detail.Since the Poincare map has com-plex properties including single hump and multiple-hump functions and multiple discontinuous points,which can result in the coexistence of multiple order-1 period-ic solutions.We develop some new analytical methods and techniques of nonlinear state dependent impulsive system.Our research methods develop and enrich the theoretical analysis methods and techniques of nonlinear state dependent impulsive system,which could be used as the basis for the pest control strategy.In practice,if the number of pest is large or natural enemies is small in the fields,the larger number of natural enemies should be released and vice versa,i.e.the nonlinear releasing function is a increasing function of the number of pest and a decreasing function of the number of natural enemies.In chapter 5,in view of the nonlinear releasing function,a Holling ? predator-prey model with nonlinear state-dependent feedback control is developed and investigated.The existence and stability of the semi-trivial periodic solution are provided by the analogue of the analogue of Poincare criterion.Further,bifurcation theorems related to the discrete map are used to address the bifurcations of a semi-trivial periodic solution.By selecting control parameters and system parameters,the transcritical bifurcations and pitchfork bifurcations are provided.